33. Diagonalization

Try these first

1. 1.

   What is the condition for a matrix to be diagonalizable?

   Hint

   Consider the relationship between eigenvalues and the independence of eigenvectors.

2. 2.

   Define eigenvalue in your own words.

   Hint

   Think about what happens to vectors when transformed by matrices.

3. 3.

   What does it mean for a matrix to be diagonalizable?

   • It can be reduced to a zero matrix
   • It has a complete set of eigenvectors
   • It is always symmetric

   Hint

   Think of the properties of the eigenvectors.

4. 4.

   True or False: All matrices with repeated eigenvalues are non-diagonalizable.

   • True
   • False

   Hint

   Consider examples of matrices with distinct eigenvalues.

5. 5.

   Given the matrix A = [2 3; 2 4], demonstrate if A is diagonalizable and find its eigenvalues and eigenvectors.

   Hint

   Focus on the linearly independent nature of the eigenvectors.

6. 6.

   Prove the Jordan form is useful for non-diagonalizable matrices by providing an example and explaining the process.

   Hint

   Think about the roles of blocks in Jordan form.